Question

3. Alice and Bob are practicing their basketball free throws. Alice sinks it with probability a > 0 on each try, Bob is successful with probability b > 0 each time. The outcomes of their attempts are all independent from each other. Let A be the number of Alice’s tries until her first success. Let B be the same for Bob. Let C be equal to 1 if Alice gets to score with fewer attempts, 2 if Bob gets to score with fewer attempts, and 3 if they both need the same number of attempts to score. Finally, let D := min{A, B}, that is, the number of attempts of the first successful player, whoever that may be.

(a) Determine the probability mass function (pmf) of C.

(b) Determine the probability mass function (pmf) of D.

(c) Determine the expected value of D.

Answer 1